MA 259 - Differential Equations

1. Solve simple differential equations, including Separable, Homogeneous, Exact, First-Order Linear, homogeneous of degree zero, Bernoulli, and higher-order equations.
2. Apply differential equations to real-world problems.
3. Solve nth order Linear Differential Equations, especially those with constant coefficients.
4. Solve systems of differential equations.
5. Find Power Series solutions to differential equations.
6. Phase plane analysis with nonlinear systems by utilizing linearization at equilibrium points and by identifying the solutions behavior.
7. Demonstrate numerical methods for solving differential equations, and be able to use technology to implement these methods.
8. Define Laplace Transform; state, prove, and use properties of Laplace Transform and their inverses, and use Laplace Transform to solve differential equations. 
9. Effectively organize communications, ensuring there is a clear introduction and conclusion, the content is well sequenced, and there are appropriate transitions. 
10. Clearly and completely state and describe a problem/issue. 
11. Complete work accurately with attention to detail. 

A. Definition of Ordinary Differential Equations
B. General Solutions
C. Particular Solutions
D. Separable Equations
E. Homogeneous Equations
F. Exact Equations

G. First Order Linear Equations

H. Picard’s Method

I. Second Order Equations Reducible to First Order
J. Applications by Modeling with Differential Equations

II. Linear Differential Equations

A. Definition of nth Order Linear Differential Equations
B. Solutions of Equations with Constant Coefficients
C. Non-Constant Coefficients: Cauchy-Euler Equations
D. General Solution of Non-Homogeneous Equations
E. Method of Undetermined Coefficients
F. Variation of Parameters
G. Applications

III.  Systems of Differential Equations

A. Definition of First Order Systems
B. Solution of Systems by Elimination or Substitution
C. Representation of Systems by Matrices
D. Solution of Systems by Eigenvectors
E. Non-Homogeneous Linear Systems

F. Nonlinear Systems

G. Identification of Equilibrium Points.

H. Linearization and Phase Plane Analysis
I. Applications by Modeling with Systems of Differential Equations

IV. Series Solutions

A. Power Series
B. Taylor Series
C. Operations with Series Including Re-indexing
D. Series Solutions: Ordinary Points
E. Series Solutions: Singular Points

V. Numerical Methods

A. Euler Method
B. Runge-Kutta Methods

C. Student Project Emphasizing Numerical Methods

VI. Laplace Transform

A. Definition of Laplace Transform
B. Computing Laplace Transform
C. Properties of Laplace Transform
D. Computing Inverse Laplace Transform

E. Solving Differential Equations using Laplace Transform

F. Engineering Applications of Laplace Transform